def test_regularized_training(self):
"""Test that adding regularization penalty to the training loss works."""
np.random.seed(0)
# Set up the problem of recovering w given x and
# y = x . w + noise
# with the a priori assumption that w is sparse. There are fewer examples
# than dimensions (x is a wide matrix), so the problem is underdetermined
# without the sparsity assumption.
num_examples, num_dim = 8, 10
x = np.random.randn(num_examples, num_dim).astype(np.float32)
true_w = np.zeros((num_dim, 2), np.float32)
true_w[[2, 4, 6], 0] = [1.0, 2.0, 3.0]
true_w[[3, 5], 1] = [4.0, 5.0]
y = np.dot(x, true_w) + 1e-3 * np.random.randn(num_examples, 2)
# Get the least squares estimate for w. It isn't very accurate.
least_squares_w = np.linalg.lstsq(x, y, rcond=None)[0]
least_squares_w_error = hk_util.l2_loss(least_squares_w - true_w)
# Get a better estimate by solving the L1 regularized problem
# argmin_w ||x . w - y||_2^2 + c ||w||_1.
w_regularizer = lambda w: 4.0 * hk_util.l1_loss(w)
def model_fun(batch):
x = batch['x']
return hk_util.Linear(2, use_bias=False, w_regularizer=w_regularizer)(x)
model = hk.transform(model_fun)
def loss_fun(params, batch):
"""Training loss with L1 regularization penalty term."""
y_predicted, penalties = model.apply(params, batch)
return hk_util.l2_loss(y_predicted - batch['y']) + penalties
batch = {'x': x, 'y': y}
params = model.init(jax.random.PRNGKey(0), batch)
optimizer = optix.chain( # Gradient descent with decreasing learning rate.
optix.trace(decay=0.0, nesterov=False),
optix.scale_by_schedule(lambda i: -0.05 / jnp.sqrt(1 + i)))
opt_state = optimizer.init(params)
@jax.jit
def train_step(params, opt_state, batch):
grads = jax.grad(loss_fun)(params, batch)
updates, opt_state = optimizer.update(grads, opt_state)
new_params = optix.apply_updates(params, updates)
return new_params, opt_state
for _ in range(1000):
params, opt_state = train_step(params, opt_state, batch)
l1_w = params['linear']['w']
l1_w_error = hk_util.l2_loss(l1_w - true_w).item()
# The L1-regularized estimate is much more accurate.
self.assertGreater(least_squares_w_error, 4.0)
self.assertLess(l1_w_error, 1.0)
def make_optimizer():
"""SGD with nesterov momentum and a custom lr schedule."""
return optix.chain(
optix.trace(decay=FLAGS.optimizer_momentum,
nesterov=FLAGS.optimizer_use_nesterov),
optix.scale_by_schedule(lr_schedule), optix.scale(-1))
def make_optimizer(lr_schedule, momentum_decay):
return optix.chain(optix.trace(decay=momentum_decay, nesterov=False),
optix.scale_by_schedule(lr_schedule),
optix.scale(-1))
def make_optimizer(lr_schedule, momentum_decay):
"""Make SGD optimizer with momentum."""
# Maximize log-prob instead of minimizing loss
return optix.chain(optix.trace(decay=momentum_decay, nesterov=False),
optix.scale_by_schedule(lr_schedule))